#6867478

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,22,27,23,28,21,26)(19,24,32,40,30,38,31,39)(20,33,34,41,35,42,36,25)(29,43,45,47,44,48,46,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,30)(18,24,28,38)(20,29,35,44)(21,31,22,32)(25,37,41,47)(26,39,27,40)(33,43,42,48)(34,45,36,46);(10,14)(15,16)(17,20,23,35)(18,25,28,41)(19,29,30,44)(21,34,22,36)(24,37,38,47)(26,33,27,42)(31,45,32,46)(39,43,40,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,30)(24,38)(29,44)(31,32)(37,47)(39,40)(43,48)(45,46);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,28,27)(19,31,30,32)(20,34,35,36)(24,39,38,40)(25,33,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6774674);